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lift_ι (W : C) (k : Π a, W ⟶ I.left a)
(h : ∀ b, k (I.fst_to b) ≫ I.fst b = k (I.snd_to b) ≫ I.snd b) (a) :
multiequalizer.lift I _ k h ≫ multiequalizer.ι I a = k _ | limit.lift_π _ _ | lemma | category_theory.limits.multiequalizer.lift_ι | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_ext {W : C} (i j : W ⟶ multiequalizer I)
(h : ∀ a, i ≫ multiequalizer.ι I a =
j ≫ multiequalizer.ι I a) :
i = j | limit.hom_ext
begin
rintro (a|b),
{ apply h },
simp_rw [← limit.w I.multicospan (walking_multicospan.hom.fst b),
← category.assoc, h],
end | lemma | category_theory.limits.multiequalizer.hom_ext | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [
"hom_ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_equalizer : multiequalizer I ≅ equalizer I.fst_pi_map I.snd_pi_map | limit.iso_limit_cone ⟨_, is_limit.of_preserves_cone_terminal
I.multifork_equiv_pi_fork.inverse (limit.is_limit _)⟩ | def | category_theory.limits.multiequalizer.iso_equalizer | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | The multiequalizer is isomorphic to the equalizer of `∏ I.left ⇉ ∏ I.right`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_pi : multiequalizer I ⟶ ∏ I.left | (iso_equalizer I).hom ≫ equalizer.ι I.fst_pi_map I.snd_pi_map | def | category_theory.limits.multiequalizer.ι_pi | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | The canonical injection `multiequalizer I ⟶ ∏ I.left`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_pi_π (a) : ι_pi I ≫ pi.π I.left a = ι I a | by { rw [ι_pi, category.assoc, ← iso.eq_inv_comp, iso_equalizer], simpa } | lemma | category_theory.limits.multiequalizer.ι_pi_π | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
π (b : I.R) : I.right b ⟶ multicoequalizer I | colimit.ι I.multispan (walking_multispan.right _) | abbreviation | category_theory.limits.multicoequalizer.π | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | The canonical map from the multiequalizer to the objects on the left. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
multicofork : multicofork I | colimit.cocone _ | abbreviation | category_theory.limits.multicoequalizer.multicofork | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | The multicofork associated to the multicoequalizer. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
multicofork_π (b) :
(multicoequalizer.multicofork I).π b = multicoequalizer.π I b | rfl | lemma | category_theory.limits.multicoequalizer.multicofork_π | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
multicofork_ι_app_right (b) :
(multicoequalizer.multicofork I).ι.app (walking_multispan.right b) =
multicoequalizer.π I b | rfl | lemma | category_theory.limits.multicoequalizer.multicofork_ι_app_right | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
condition (a) :
I.fst a ≫ multicoequalizer.π I (I.fst_from a) =
I.snd a ≫ multicoequalizer.π I (I.snd_from a) | multicofork.condition _ _ | lemma | category_theory.limits.multicoequalizer.condition | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
desc (W : C) (k : Π b, I.right b ⟶ W)
(h : ∀ a, I.fst a ≫ k (I.fst_from a) = I.snd a ≫ k (I.snd_from a)) :
multicoequalizer I ⟶ W | colimit.desc _ (multicofork.of_π I _ k h) | abbreviation | category_theory.limits.multicoequalizer.desc | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | Construct a morphism from the multicoequalizer from its universal property. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
π_desc (W : C) (k : Π b, I.right b ⟶ W)
(h : ∀ a, I.fst a ≫ k (I.fst_from a) = I.snd a ≫ k (I.snd_from a)) (b) :
multicoequalizer.π I b ≫ multicoequalizer.desc I _ k h = k _ | colimit.ι_desc _ _ | lemma | category_theory.limits.multicoequalizer.π_desc | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_ext {W : C} (i j : multicoequalizer I ⟶ W)
(h : ∀ b, multicoequalizer.π I b ≫ i = multicoequalizer.π I b ≫ j) :
i = j | colimit.hom_ext
begin
rintro (a|b),
{ simp_rw [← colimit.w I.multispan (walking_multispan.hom.fst a),
category.assoc, h] },
{ apply h },
end | lemma | category_theory.limits.multicoequalizer.hom_ext | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [
"hom_ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso_coequalizer : multicoequalizer I ≅ coequalizer I.fst_sigma_map I.snd_sigma_map | colimit.iso_colimit_cocone ⟨_, is_colimit.of_preserves_cocone_initial
I.multicofork_equiv_sigma_cofork.inverse (colimit.is_colimit _)⟩ | def | category_theory.limits.multicoequalizer.iso_coequalizer | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | The multicoequalizer is isomorphic to the coequalizer of `∐ I.left ⇉ ∐ I.right`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sigma_π : ∐ I.right ⟶ multicoequalizer I | coequalizer.π I.fst_sigma_map I.snd_sigma_map ≫ (iso_coequalizer I).inv | def | category_theory.limits.multicoequalizer.sigma_π | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | The canonical projection `∐ I.right ⟶ multicoequalizer I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_sigma_π (b) : sigma.ι I.right b ≫ sigma_π I = π I b | by { rw [sigma_π, ← category.assoc, iso.comp_inv_eq, iso_coequalizer], simpa } | lemma | category_theory.limits.multicoequalizer.ι_sigma_π | category_theory.limits.shapes | src/category_theory/limits/shapes/multiequalizer.lean | [
"category_theory.limits.shapes.products",
"category_theory.limits.shapes.equalizers",
"category_theory.limits.cone_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fan (f : β → C) | cone (discrete.functor f) | abbreviation | category_theory.limits.fan | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | A fan over `f : β → C` consists of a collection of maps from an object `P` to every `f b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cofan (f : β → C) | cocone (discrete.functor f) | abbreviation | category_theory.limits.cofan | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | A cofan over `f : β → C` consists of a collection of maps from every `f b` to an object `P`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fan.mk {f : β → C} (P : C) (p : Π b, P ⟶ f b) : fan f | { X := P,
π := { app := λ X, p X.as } } | def | category_theory.limits.fan.mk | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | A fan over `f : β → C` consists of a collection of maps from an object `P` to every `f b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cofan.mk {f : β → C} (P : C) (p : Π b, f b ⟶ P) : cofan f | { X := P,
ι := { app := λ X, p X.as } } | def | category_theory.limits.cofan.mk | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | A cofan over `f : β → C` consists of a collection of maps from every `f b` to an object `P`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fan.proj {f : β → C} (p : fan f) (j : β) : p.X ⟶ f j | p.π.app (discrete.mk j) | def | category_theory.limits.fan.proj | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | Get the `j`th map in the fan | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fan_mk_proj {f : β → C} (P : C) (p : Π b, P ⟶ f b) (j : β) :
(fan.mk P p).proj j = p j | rfl | lemma | category_theory.limits.fan_mk_proj | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_product (f : β → C) | has_limit (discrete.functor f) | abbreviation | category_theory.limits.has_product | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | An abbreviation for `has_limit (discrete.functor f)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_coproduct (f : β → C) | has_colimit (discrete.functor f) | abbreviation | category_theory.limits.has_coproduct | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | An abbreviation for `has_colimit (discrete.functor f)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_fan_limit {f : β → C} (t : fan f)
(lift : Π s : fan f, s.X ⟶ t.X)
(fac : ∀ (s : fan f) (j : β), lift s ≫ (t.proj j) = s.proj j)
(uniq : ∀ (s : fan f) (m : s.X ⟶ t.X) (w : ∀ j : β, m ≫ t.proj j = s.proj j), m = lift s) :
is_limit t | { lift := lift,
fac' := λ s j, by convert fac s j.as; simp,
uniq' := λ s m w, uniq s m (λ j, w (discrete.mk j)), } | def | category_theory.limits.mk_fan_limit | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [
"lift"
] | Make a fan `f` into a limit fan by providing `lift`, `fac`, and `uniq` --
just a convenience lemma to avoid having to go through `discrete` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_products_of_shape (β : Type v) | has_limits_of_shape.{v} (discrete β) | abbreviation | category_theory.limits.has_products_of_shape | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | An abbreviation for `has_limits_of_shape (discrete f)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_coproducts_of_shape (β : Type v) | has_colimits_of_shape.{v} (discrete β) | abbreviation | category_theory.limits.has_coproducts_of_shape | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | An abbreviation for `has_colimits_of_shape (discrete f)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi_obj (f : β → C) [has_product f] | limit (discrete.functor f) | abbreviation | category_theory.limits.pi_obj | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | `pi_obj f` computes the product of a family of elements `f`.
(It is defined as an abbreviation for `limit (discrete.functor f)`,
so for most facts about `pi_obj f`, you will just use general facts about limits.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sigma_obj (f : β → C) [has_coproduct f] | colimit (discrete.functor f) | abbreviation | category_theory.limits.sigma_obj | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | `sigma_obj f` computes the coproduct of a family of elements `f`.
(It is defined as an abbreviation for `colimit (discrete.functor f)`,
so for most facts about `sigma_obj f`, you will just use general facts about colimits.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi.π (f : β → C) [has_product f] (b : β) : ∏ f ⟶ f b | limit.π (discrete.functor f) (discrete.mk b) | abbreviation | category_theory.limits.pi.π | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | The `b`-th projection from the pi object over `f` has the form `∏ f ⟶ f b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sigma.ι (f : β → C) [has_coproduct f] (b : β) : f b ⟶ ∐ f | colimit.ι (discrete.functor f) (discrete.mk b) | abbreviation | category_theory.limits.sigma.ι | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | The `b`-th inclusion into the sigma object over `f` has the form `f b ⟶ ∐ f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
product_is_product (f : β → C) [has_product f] :
is_limit (fan.mk _ (pi.π f)) | is_limit.of_iso_limit (limit.is_limit (discrete.functor f)) (cones.ext (iso.refl _) (by tidy)) | def | category_theory.limits.product_is_product | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | The fan constructed of the projections from the product is limiting. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coproduct_is_coproduct (f : β → C) [has_coproduct f] :
is_colimit (cofan.mk _ (sigma.ι f)) | is_colimit.of_iso_colimit (colimit.is_colimit (discrete.functor f)) (cocones.ext (iso.refl _)
(by tidy)) | def | category_theory.limits.coproduct_is_coproduct | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | The cofan constructed of the inclusions from the coproduct is colimiting. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi.lift {f : β → C} [has_product f] {P : C} (p : Π b, P ⟶ f b) : P ⟶ ∏ f | limit.lift _ (fan.mk P p) | abbreviation | category_theory.limits.pi.lift | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | A collection of morphisms `P ⟶ f b` induces a morphism `P ⟶ ∏ f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sigma.desc {f : β → C} [has_coproduct f] {P : C} (p : Π b, f b ⟶ P) : ∐ f ⟶ P | colimit.desc _ (cofan.mk P p) | abbreviation | category_theory.limits.sigma.desc | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | A collection of morphisms `f b ⟶ P` induces a morphism `∐ f ⟶ P`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi.map {f g : β → C} [has_product f] [has_product g]
(p : Π b, f b ⟶ g b) : ∏ f ⟶ ∏ g | lim_map (discrete.nat_trans (λ X, p X.as)) | abbreviation | category_theory.limits.pi.map | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | Construct a morphism between categorical products (indexed by the same type)
from a family of morphisms between the factors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi.map_mono {f g : β → C} [has_product f] [has_product g]
(p : Π b, f b ⟶ g b) [Π i, mono (p i)] : mono $ pi.map p | @@limits.lim_map_mono _ _ _ _ _ (by { dsimp, apply_instance }) | instance | category_theory.limits.pi.map_mono | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.map_iso {f g : β → C} [has_products_of_shape β C]
(p : Π b, f b ≅ g b) : ∏ f ≅ ∏ g | lim.map_iso (discrete.nat_iso (λ X, p X.as)) | abbreviation | category_theory.limits.pi.map_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | Construct an isomorphism between categorical products (indexed by the same type)
from a family of isomorphisms between the factors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sigma.map {f g : β → C} [has_coproduct f] [has_coproduct g]
(p : Π b, f b ⟶ g b) : ∐ f ⟶ ∐ g | colim_map (discrete.nat_trans (λ X, p X.as)) | abbreviation | category_theory.limits.sigma.map | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [
"sigma.map"
] | Construct a morphism between categorical coproducts (indexed by the same type)
from a family of morphisms between the factors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sigma.map_epi {f g : β → C} [has_coproduct f] [has_coproduct g]
(p : Π b, f b ⟶ g b) [Π i, epi (p i)] : epi $ sigma.map p | @@limits.colim_map_epi _ _ _ _ _ (by { dsimp, apply_instance }) | instance | category_theory.limits.sigma.map_epi | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [
"sigma.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sigma.map_iso {f g : β → C} [has_coproducts_of_shape β C]
(p : Π b, f b ≅ g b) : ∐ f ≅ ∐ g | colim.map_iso (discrete.nat_iso (λ X, p X.as)) | abbreviation | category_theory.limits.sigma.map_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | Construct an isomorphism between categorical coproducts (indexed by the same type)
from a family of isomorphisms between the factors. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi_comparison [has_product f] [has_product (λ b, G.obj (f b))] :
G.obj (∏ f) ⟶ ∏ (λ b, G.obj (f b)) | pi.lift (λ b, G.map (pi.π f b)) | def | category_theory.limits.pi_comparison | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | The comparison morphism for the product of `f`. This is an iso iff `G` preserves the product
of `f`, see `preserves_product.of_iso_comparison`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi_comparison_comp_π [has_product f] [has_product (λ b, G.obj (f b))] (b : β) :
pi_comparison G f ≫ pi.π _ b = G.map (pi.π f b) | limit.lift_π _ (discrete.mk b) | lemma | category_theory.limits.pi_comparison_comp_π | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_lift_pi_comparison [has_product f] [has_product (λ b, G.obj (f b))]
(P : C) (g : Π j, P ⟶ f j) :
G.map (pi.lift g) ≫ pi_comparison G f = pi.lift (λ j, G.map (g j)) | by { ext, discrete_cases, simp [← G.map_comp] } | lemma | category_theory.limits.map_lift_pi_comparison | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sigma_comparison [has_coproduct f] [has_coproduct (λ b, G.obj (f b))] :
∐ (λ b, G.obj (f b)) ⟶ G.obj (∐ f) | sigma.desc (λ b, G.map (sigma.ι f b)) | def | category_theory.limits.sigma_comparison | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | The comparison morphism for the coproduct of `f`. This is an iso iff `G` preserves the coproduct
of `f`, see `preserves_coproduct.of_iso_comparison`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_comp_sigma_comparison [has_coproduct f] [has_coproduct (λ b, G.obj (f b))] (b : β) :
sigma.ι _ b ≫ sigma_comparison G f = G.map (sigma.ι f b) | colimit.ι_desc _ (discrete.mk b) | lemma | category_theory.limits.ι_comp_sigma_comparison | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sigma_comparison_map_desc [has_coproduct f] [has_coproduct (λ b, G.obj (f b))]
(P : C) (g : Π j, f j ⟶ P) :
sigma_comparison G f ≫ G.map (sigma.desc g) = sigma.desc (λ j, G.map (g j)) | by { ext, discrete_cases, simp [← G.map_comp] } | lemma | category_theory.limits.sigma_comparison_map_desc | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_products | Π (J : Type w), has_limits_of_shape (discrete J) C | abbreviation | category_theory.limits.has_products | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | An abbreviation for `Π J, has_limits_of_shape (discrete J) C` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_coproducts | Π (J : Type w), has_colimits_of_shape (discrete J) C | abbreviation | category_theory.limits.has_coproducts | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | An abbreviation for `Π J, has_colimits_of_shape (discrete J) C` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_smallest_products_of_has_products [has_products.{w} C] : has_products.{0} C | λ J, has_limits_of_shape_of_equivalence
(discrete.equivalence equiv.ulift : discrete (ulift.{w} J) ≌ _) | lemma | category_theory.limits.has_smallest_products_of_has_products | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [
"equiv.ulift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_smallest_coproducts_of_has_coproducts [has_coproducts.{w} C] : has_coproducts.{0} C | λ J, has_colimits_of_shape_of_equivalence
(discrete.equivalence equiv.ulift : discrete (ulift.{w} J) ≌ _) | lemma | category_theory.limits.has_smallest_coproducts_of_has_coproducts | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [
"equiv.ulift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_products_of_limit_fans (lf : ∀ {J : Type w} (f : J → C), fan f)
(lf_is_limit : ∀ {J : Type w} (f : J → C), is_limit (lf f)) : has_products.{w} C | λ (J : Type w), { has_limit := λ F, has_limit.mk
⟨(cones.postcompose discrete.nat_iso_functor.inv).obj (lf (λ j, F.obj ⟨j⟩)),
(is_limit.postcompose_inv_equiv _ _).symm (lf_is_limit _)⟩ } | lemma | category_theory.limits.has_products_of_limit_fans | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_cone_of_unique : limit_cone (discrete.functor f) | { cone :=
{ X := f default,
π := { app := λ j, eq_to_hom (by { dsimp, congr, }), }, },
is_limit :=
{ lift := λ s, s.π.app default,
fac' := λ s j, begin
have w := (s.π.naturality (eq_to_hom (unique.default_eq _))).symm,
dsimp at w,
simpa [eq_to_hom_map] using w,
end,
uniq' := λ s ... | def | category_theory.limits.limit_cone_of_unique | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [
"lift",
"unique.default_eq"
] | The limit cone for the product over an index type with exactly one term. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_product_unique : has_product f | has_limit.mk (limit_cone_of_unique f) | instance | category_theory.limits.has_product_unique | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
product_unique_iso : ∏ f ≅ f default | is_limit.cone_point_unique_up_to_iso (limit.is_limit _) (limit_cone_of_unique f).is_limit | def | category_theory.limits.product_unique_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | A product over a index type with exactly one term is just the object over that term. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
colimit_cocone_of_unique : colimit_cocone (discrete.functor f) | { cocone :=
{ X := f default,
ι := { app := λ j, eq_to_hom (by { discrete_cases, dsimp, congr, }), }, },
is_colimit :=
{ desc := λ s, s.ι.app default,
fac' := λ s j, begin
have w := (s.ι.naturality (eq_to_hom (unique.eq_default _))),
dsimp at w,
simpa [eq_to_hom_map] using w,
end,
... | def | category_theory.limits.colimit_cocone_of_unique | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [
"unique.eq_default"
] | The colimit cocone for the coproduct over an index type with exactly one term. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_coproduct_unique : has_coproduct f | has_colimit.mk (colimit_cocone_of_unique f) | instance | category_theory.limits.has_coproduct_unique | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coproduct_unique_iso : ∐ f ≅ f default | is_colimit.cocone_point_unique_up_to_iso (colimit.is_colimit _)
(colimit_cocone_of_unique f).is_colimit | def | category_theory.limits.coproduct_unique_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | A coproduct over a index type with exactly one term is just the object over that term. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi.reindex : pi_obj (f ∘ ε) ≅ pi_obj f | has_limit.iso_of_equivalence (discrete.equivalence ε) (discrete.nat_iso (λ i, iso.refl _)) | def | category_theory.limits.pi.reindex | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | Reindex a categorical product via an equivalence of the index types. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi.reindex_hom_π (b : β) : (pi.reindex ε f).hom ≫ pi.π f (ε b) = pi.π (f ∘ ε) b | begin
dsimp [pi.reindex],
simp only [has_limit.iso_of_equivalence_hom_π, discrete.nat_iso_inv_app,
equivalence.equivalence_mk'_counit, discrete.equivalence_counit_iso, discrete.nat_iso_hom_app,
eq_to_iso.hom, eq_to_hom_map],
dsimp,
simpa [eq_to_hom_map] using
limit.w (discrete.functor (f ∘ ε)) (disc... | lemma | category_theory.limits.pi.reindex_hom_π | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.reindex_inv_π (b : β) : (pi.reindex ε f).inv ≫ pi.π (f ∘ ε) b = pi.π f (ε b) | by simp [iso.inv_comp_eq] | lemma | category_theory.limits.pi.reindex_inv_π | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sigma.reindex : sigma_obj (f ∘ ε) ≅ sigma_obj f | has_colimit.iso_of_equivalence (discrete.equivalence ε) (discrete.nat_iso (λ i, iso.refl _)) | def | category_theory.limits.sigma.reindex | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | Reindex a categorical coproduct via an equivalence of the index types. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sigma.ι_reindex_hom (b : β) : sigma.ι (f ∘ ε) b ≫ (sigma.reindex ε f).hom = sigma.ι f (ε b) | begin
dsimp [sigma.reindex],
simp only [has_colimit.iso_of_equivalence_hom_π, equivalence.equivalence_mk'_unit,
discrete.equivalence_unit_iso, discrete.nat_iso_hom_app, eq_to_iso.hom, eq_to_hom_map,
discrete.nat_iso_inv_app],
dsimp,
simp [eq_to_hom_map,
←colimit.w (discrete.functor f) (discrete.eq_t... | lemma | category_theory.limits.sigma.ι_reindex_hom | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sigma.ι_reindex_inv (b : β) : sigma.ι f (ε b) ≫ (sigma.reindex ε f).inv = sigma.ι (f ∘ ε) b | by simp [iso.comp_inv_eq] | lemma | category_theory.limits.sigma.ι_reindex_inv | category_theory.limits.shapes | src/category_theory/limits/shapes/products.lean | [
"category_theory.limits.has_limits",
"category_theory.discrete_category"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
walking_cospan : Type | wide_pullback_shape walking_pair | abbreviation | category_theory.limits.walking_cospan | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | The type of objects for the diagram indexing a pullback, defined as a special case of
`wide_pullback_shape`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
walking_cospan.left : walking_cospan | some walking_pair.left | abbreviation | category_theory.limits.walking_cospan.left | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | The left point of the walking cospan. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
walking_cospan.right : walking_cospan | some walking_pair.right | abbreviation | category_theory.limits.walking_cospan.right | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | The right point of the walking cospan. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
walking_cospan.one : walking_cospan | none | abbreviation | category_theory.limits.walking_cospan.one | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | The central point of the walking cospan. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
walking_span : Type | wide_pushout_shape walking_pair | abbreviation | category_theory.limits.walking_span | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | The type of objects for the diagram indexing a pushout, defined as a special case of
`wide_pushout_shape`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
walking_span.left : walking_span | some walking_pair.left | abbreviation | category_theory.limits.walking_span.left | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | The left point of the walking span. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
walking_span.right : walking_span | some walking_pair.right | abbreviation | category_theory.limits.walking_span.right | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | The right point of the walking span. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
walking_span.zero : walking_span | none | abbreviation | category_theory.limits.walking_span.zero | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | The central point of the walking span. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom : walking_cospan → walking_cospan → Type | wide_pullback_shape.hom | abbreviation | category_theory.limits.walking_cospan.hom | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | The type of arrows for the diagram indexing a pullback. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom.inl : left ⟶ one | wide_pullback_shape.hom.term _ | abbreviation | category_theory.limits.walking_cospan.hom.inl | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | The left arrow of the walking cospan. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom.inr : right ⟶ one | wide_pullback_shape.hom.term _ | abbreviation | category_theory.limits.walking_cospan.hom.inr | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | The right arrow of the walking cospan. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom.id (X : walking_cospan) : X ⟶ X | wide_pullback_shape.hom.id X | abbreviation | category_theory.limits.walking_cospan.hom.id | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | The identity arrows of the walking cospan. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom : walking_span → walking_span → Type | wide_pushout_shape.hom | abbreviation | category_theory.limits.walking_span.hom | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | The type of arrows for the diagram indexing a pushout. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom.fst : zero ⟶ left | wide_pushout_shape.hom.init _ | abbreviation | category_theory.limits.walking_span.hom.fst | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | The left arrow of the walking span. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom.snd : zero ⟶ right | wide_pushout_shape.hom.init _ | abbreviation | category_theory.limits.walking_span.hom.snd | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | The right arrow of the walking span. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom.id (X : walking_span) : X ⟶ X | wide_pushout_shape.hom.id X | abbreviation | category_theory.limits.walking_span.hom.id | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | The identity arrows of the walking span. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
walking_cospan.ext {F : walking_cospan ⥤ C} {s t : cone F} (i : s.X ≅ t.X)
(w₁ : s.π.app walking_cospan.left = i.hom ≫ t.π.app walking_cospan.left)
(w₂ : s.π.app walking_cospan.right = i.hom ≫ t.π.app walking_cospan.right) :
s ≅ t | begin
apply cones.ext i,
rintro (⟨⟩|⟨⟨⟩⟩),
{ have h₁ := s.π.naturality walking_cospan.hom.inl,
dsimp at h₁, simp only [category.id_comp] at h₁,
have h₂ := t.π.naturality walking_cospan.hom.inl,
dsimp at h₂, simp only [category.id_comp] at h₂,
simp_rw [h₂, ←category.assoc, ←w₁, ←h₁], },
{ exact w... | def | category_theory.limits.walking_cospan.ext | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | To construct an isomorphism of cones over the walking cospan,
it suffices to construct an isomorphism
of the cone points and check it commutes with the legs to `left` and `right`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
walking_span.ext {F : walking_span ⥤ C} {s t : cocone F} (i : s.X ≅ t.X)
(w₁ : s.ι.app walking_cospan.left ≫ i.hom = t.ι.app walking_cospan.left)
(w₂ : s.ι.app walking_cospan.right ≫ i.hom = t.ι.app walking_cospan.right) :
s ≅ t | begin
apply cocones.ext i,
rintro (⟨⟩|⟨⟨⟩⟩),
{ have h₁ := s.ι.naturality walking_span.hom.fst,
dsimp at h₁, simp only [category.comp_id] at h₁,
have h₂ := t.ι.naturality walking_span.hom.fst,
dsimp at h₂, simp only [category.comp_id] at h₂,
simp_rw [←h₁, category.assoc, w₁, h₂], },
{ exact w₁, }... | def | category_theory.limits.walking_span.ext | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | To construct an isomorphism of cocones over the walking span,
it suffices to construct an isomorphism
of the cocone points and check it commutes with the legs from `left` and `right`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cospan {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) : walking_cospan ⥤ C | wide_pullback_shape.wide_cospan Z
(λ j, walking_pair.cases_on j X Y) (λ j, walking_pair.cases_on j f g) | def | category_theory.limits.cospan | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | `cospan f g` is the functor from the walking cospan hitting `f` and `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
span {X Y Z : C} (f : X ⟶ Y) (g : X ⟶ Z) : walking_span ⥤ C | wide_pushout_shape.wide_span X
(λ j, walking_pair.cases_on j Y Z) (λ j, walking_pair.cases_on j f g) | def | category_theory.limits.span | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | `span f g` is the functor from the walking span hitting `f` and `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cospan_left {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) :
(cospan f g).obj walking_cospan.left = X | rfl | lemma | category_theory.limits.cospan_left | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_left {X Y Z : C} (f : X ⟶ Y) (g : X ⟶ Z) :
(span f g).obj walking_span.left = Y | rfl | lemma | category_theory.limits.span_left | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cospan_right {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) :
(cospan f g).obj walking_cospan.right = Y | rfl | lemma | category_theory.limits.cospan_right | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_right {X Y Z : C} (f : X ⟶ Y) (g : X ⟶ Z) :
(span f g).obj walking_span.right = Z | rfl | lemma | category_theory.limits.span_right | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cospan_one {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) :
(cospan f g).obj walking_cospan.one = Z | rfl | lemma | category_theory.limits.cospan_one | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_zero {X Y Z : C} (f : X ⟶ Y) (g : X ⟶ Z) :
(span f g).obj walking_span.zero = X | rfl | lemma | category_theory.limits.span_zero | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cospan_map_inl {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) :
(cospan f g).map walking_cospan.hom.inl = f | rfl | lemma | category_theory.limits.cospan_map_inl | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_map_fst {X Y Z : C} (f : X ⟶ Y) (g : X ⟶ Z) :
(span f g).map walking_span.hom.fst = f | rfl | lemma | category_theory.limits.span_map_fst | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cospan_map_inr {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) :
(cospan f g).map walking_cospan.hom.inr = g | rfl | lemma | category_theory.limits.cospan_map_inr | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_map_snd {X Y Z : C} (f : X ⟶ Y) (g : X ⟶ Z) :
(span f g).map walking_span.hom.snd = g | rfl | lemma | category_theory.limits.span_map_snd | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cospan_map_id {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) (w : walking_cospan) :
(cospan f g).map (walking_cospan.hom.id w) = 𝟙 _ | rfl | lemma | category_theory.limits.cospan_map_id | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_map_id {X Y Z : C} (f : X ⟶ Y) (g : X ⟶ Z) (w : walking_span) :
(span f g).map (walking_span.hom.id w) = 𝟙 _ | rfl | lemma | category_theory.limits.span_map_id | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagram_iso_cospan (F : walking_cospan ⥤ C) :
F ≅ cospan (F.map inl) (F.map inr) | nat_iso.of_components (λ j, eq_to_iso (by tidy)) (by tidy) | def | category_theory.limits.diagram_iso_cospan | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | Every diagram indexing an pullback is naturally isomorphic (actually, equal) to a `cospan` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diagram_iso_span (F : walking_span ⥤ C) :
F ≅ span (F.map fst) (F.map snd) | nat_iso.of_components (λ j, eq_to_iso (by tidy)) (by tidy) | def | category_theory.limits.diagram_iso_span | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | Every diagram indexing a pushout is naturally isomorphic (actually, equal) to a `span` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cospan_comp_iso (F : C ⥤ D) {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) :
cospan f g ⋙ F ≅ cospan (F.map f) (F.map g) | nat_iso.of_components (by rintros (⟨⟩|⟨⟨⟩⟩); exact iso.refl _)
(by rintros (⟨⟩|⟨⟨⟩⟩) (⟨⟩|⟨⟨⟩⟩) ⟨⟩; repeat { dsimp, simp, }) | def | category_theory.limits.cospan_comp_iso | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | A functor applied to a cospan is a cospan. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cospan_comp_iso_app_left :
(cospan_comp_iso F f g).app walking_cospan.left = iso.refl _ | rfl | lemma | category_theory.limits.cospan_comp_iso_app_left | category_theory.limits.shapes | src/category_theory/limits/shapes/pullbacks.lean | [
"category_theory.limits.shapes.wide_pullbacks",
"category_theory.limits.shapes.binary_products"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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